To determine which expression is equivalent to [tex]\( b^m \cdot b^m \)[/tex], let's proceed step by step by applying the properties of exponents.
1. Recall the properties of exponents:
- One important property is that when you multiply two exponential expressions with the same base, you add their exponents. Specifically:
[tex]\[
a^m \cdot a^n = a^{m+n}
\][/tex]
2. Apply this property to the given expression:
- Here, both exponential expressions have the same base [tex]\( b \)[/tex], and the exponents are both [tex]\( m \)[/tex].
3. Use the property of exponents:
- [tex]\[
b^m \cdot b^m = b^{m + m}
\][/tex]
4. Simplify the exponent:
- [tex]\[
b^{m + m} = b^{2m}
\][/tex]
5. Match the simplified expression with the options given:
- None of the options exactly match [tex]\( b^{2m} \)[/tex] directly. Let’s consider whether the options make sense when the variables are compared properly.
- Option A: [tex]\( b^{m-n} \)[/tex] – This does not match because it involves subtraction.
- Option B: [tex]\( b^{m \div n} \)[/tex] – This does not match because it involves division.
- Option C: [tex]\( b^{m \cdot n} \)[/tex] – This does not match because it involves multiplication of exponents.
- Option D: [tex]\( b^{m+n} \)[/tex] – This can match [tex]\( b^{2m} \)[/tex] if [tex]\( m = n \)[/tex]. In the case where [tex]\( m = n \)[/tex], [tex]\( m+n \)[/tex] would be equivalent to [tex]\( 2m \)[/tex].
Given that the choices include an assumption where [tex]\( m \)[/tex] and [tex]\( n \)[/tex] could be equal, the correct match for [tex]\( b^m \cdot b^m \)[/tex] under the given constraints would be:
D. [tex]\( b^{m+n} \)[/tex] with [tex]\( m = n \)[/tex].
So, the final correct answer is:
[tex]\[
\boxed{D}
\][/tex]
